Q: Recently bought an Imac desktop and every time i get texts, they also appear on my iMac. I was wondering if anyone can tell me the steps on how i can disable this option.

Text from where?  Where are they appearing on your iMac?

http://support.apple.com/kb/HT2477  Mac Basics: The essentials

New to Macs/My First Mac

Are the texts appearing in the Finder or in an app on your iMac?  If the latter which app?

Care to share which OS you are using?

Will have a HOST move your post to the Message forum.

If you are logged into both with the same iMessages account, then they will be on both devices.

I don't know how to disable that besides removing the iMessage account or disabling it. In Messages Preferences on the iMac, you can select the Accounts tab, then your iMessages account, then uncheck the Enable checkbox.